With a large-scale Monte Carlo simulation, the aging of the two-dimensional dynamic Ising and XY models is numerically studied. Starting from a disordered state, the systems are quenched to critical temperature Tc and then evolve according to dynamics of model A. By calculating the autocorrelation function A(t,t′) we observe the aging phenomenon for both spin systems. Then, relatively accurate values of the dynamic critical exponent λc and z are estimated. Especially, we confirm that for the two-dimensional XY model, there is a strong logarithmic correction to scaling in the dynamic relaxation starting from a completely disordered.